Space-Time Duality in Relativistic Diffusion via Subordination

Abstract

The Cattaneo-Vernotte model and the relativistic Schrödinger operator represent two fundamental frameworks for the relativistic modifications of normal diffusion, leading to the telegraph equation and a class of spatially nonlocal diffusion equations, respectively. This paper investigates the intrinsic connection between these two relativistic diffusion models. While it is well-established that spatially non-local diffusion arises from subordinating normal diffusion, we reveal a novel reciprocal mechanism: the normal diffusion process can be recovered by subordinating the telegraph process via the corresponding inverse subordinator. Consequently, leveraging the duality between the subordinator and the inverse subordinator, we establish a distinct duality between the telegraph equation and the spatially nonlocal diffusion equation. Furthermore, this specific duality, centered on normal diffusion, is generalized to a broader class of space-time dual operator families anchored on $C_0$-semigroups.